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Find the equation of the line perpendicular to \(\displaystyle y = \frac{3 x}{2} + 6\) that contains \(\displaystyle \left( 8, \ 9\right)\).

The slope of the line is \(\displaystyle m = \frac{3}{2}\) so the perpendicular line has slope \(\displaystyle m=- \frac{2}{3}\). The equation has the form \(\displaystyle y=- \frac{2}{3}x+b\). Using the point \(\displaystyle \left( 8, \ 9\right)\) gives the equation \(\displaystyle 9=- \frac{2}{3}\left(8\right)+b\) Solving for \(\displaystyle b\) gives \(\displaystyle b = \frac{43}{3}\). The equation of the perpendicular line is \(\displaystyle y = \frac{43}{3} - \frac{2 x}{3}\).

Download \(\LaTeX\) 

\begin{question}Find the equation of the line perpendicular to $y = \frac{3 x}{2} + 6$ that contains $\left( 8, \  9\right)$. 
    \soln{9cm}{The slope of the line is $m = \frac{3}{2}$ so the perpendicular line has slope $m=- \frac{2}{3}$. The equation has the form $y=- \frac{2}{3}x+b$. Using the point $\left( 8, \  9\right)$ gives the equation $9=- \frac{2}{3}\left(8\right)+b$ Solving for $b$ gives $b = \frac{43}{3}$.  The equation of the perpendicular line is $y = \frac{43}{3} - \frac{2 x}{3}$. }

\end{question}

Download Question and Solution Environment\(\LaTeX\)
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HTML for Canvas 
<p> <p>Find the equation of the line perpendicular to  <img class="equation_image" title=" \displaystyle y = \frac{3 x}{2} + 6 " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20%5Cfrac%7B3%20x%7D%7B2%7D%20%2B%206%20" alt="LaTeX:  \displaystyle y = \frac{3 x}{2} + 6 " data-equation-content=" \displaystyle y = \frac{3 x}{2} + 6 " />  that contains  <img class="equation_image" title=" \displaystyle \left( 8, \  9\right) " src="/equation_images/%20%5Cdisplaystyle%20%5Cleft%28%208%2C%20%5C%20%209%5Cright%29%20" alt="LaTeX:  \displaystyle \left( 8, \  9\right) " data-equation-content=" \displaystyle \left( 8, \  9\right) " /> . </p> </p>
HTML for Canvas
<p> <p>The slope of the line is  <img class="equation_image" title=" \displaystyle m = \frac{3}{2} " src="/equation_images/%20%5Cdisplaystyle%20m%20%3D%20%5Cfrac%7B3%7D%7B2%7D%20" alt="LaTeX:  \displaystyle m = \frac{3}{2} " data-equation-content=" \displaystyle m = \frac{3}{2} " />  so the perpendicular line has slope  <img class="equation_image" title=" \displaystyle m=- \frac{2}{3} " src="/equation_images/%20%5Cdisplaystyle%20m%3D-%20%5Cfrac%7B2%7D%7B3%7D%20" alt="LaTeX:  \displaystyle m=- \frac{2}{3} " data-equation-content=" \displaystyle m=- \frac{2}{3} " /> . The equation has the form  <img class="equation_image" title=" \displaystyle y=- \frac{2}{3}x+b " src="/equation_images/%20%5Cdisplaystyle%20y%3D-%20%5Cfrac%7B2%7D%7B3%7Dx%2Bb%20" alt="LaTeX:  \displaystyle y=- \frac{2}{3}x+b " data-equation-content=" \displaystyle y=- \frac{2}{3}x+b " /> . Using the point  <img class="equation_image" title=" \displaystyle \left( 8, \  9\right) " src="/equation_images/%20%5Cdisplaystyle%20%5Cleft%28%208%2C%20%5C%20%209%5Cright%29%20" alt="LaTeX:  \displaystyle \left( 8, \  9\right) " data-equation-content=" \displaystyle \left( 8, \  9\right) " />  gives the equation  <img class="equation_image" title=" \displaystyle 9=- \frac{2}{3}\left(8\right)+b " src="/equation_images/%20%5Cdisplaystyle%209%3D-%20%5Cfrac%7B2%7D%7B3%7D%5Cleft%288%5Cright%29%2Bb%20" alt="LaTeX:  \displaystyle 9=- \frac{2}{3}\left(8\right)+b " data-equation-content=" \displaystyle 9=- \frac{2}{3}\left(8\right)+b " />  Solving for  <img class="equation_image" title=" \displaystyle b " src="/equation_images/%20%5Cdisplaystyle%20b%20" alt="LaTeX:  \displaystyle b " data-equation-content=" \displaystyle b " />  gives  <img class="equation_image" title=" \displaystyle b = \frac{43}{3} " src="/equation_images/%20%5Cdisplaystyle%20b%20%3D%20%5Cfrac%7B43%7D%7B3%7D%20" alt="LaTeX:  \displaystyle b = \frac{43}{3} " data-equation-content=" \displaystyle b = \frac{43}{3} " /> .  The equation of the perpendicular line is  <img class="equation_image" title=" \displaystyle y = \frac{43}{3} - \frac{2 x}{3} " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20%5Cfrac%7B43%7D%7B3%7D%20-%20%5Cfrac%7B2%20x%7D%7B3%7D%20" alt="LaTeX:  \displaystyle y = \frac{43}{3} - \frac{2 x}{3} " data-equation-content=" \displaystyle y = \frac{43}{3} - \frac{2 x}{3} " /> . </p> </p>